This note relates topics in statistical mechanics, graph theory and
combinatorics, lattice quantum field theory, super quantum mechanics
and string theory. We give a precise relation between the dimer model
on a graph embedded on a torus and the massless free Majorana fermion
living on the same lattice. A loop expansion of the fermion determinant
is performed, where the loops turn out to be compositions of two
perfect matchings. These loop states are sorted into co-chain groups
using categorification techniques similar to the ones used for categorifying
knot polynomials. The Euler characteristic of the resulting co-chain
complex recovers the Newton polynomial of the dimer model. We reinterpret
this system as supersymmetric quantum mechanics, where configurations
with vanishing net winding number form the ground states. Finally, we make use of the quiver gauge
theory–dimer model correspondence
to obtain an interpretation of the loops in terms of the physics of
D-branes probing a toric Calabi–Yau singularity.